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Evaluate int(0)^(pi)(e^(cosx))/(e^(cosx)...

Evaluate `int_(0)^(pi)(e^(cosx))/(e^(cosx)+e^(-cosx))dx`.

Text Solution

Verified by Experts

The correct Answer is:
`(pi)/(2)`
Let `I=int_(0)^(pi)(e^(cosx))/(e^(cosx)+e^(-cosx))dx` . . . (i)
`=int_(0)^(pi)(e^(cosx)(pi-x))/(e^(cos(pi-x))+e^(-cos(pi-x)))dx`
`[:' int_(0)^(a)f(x)dx = int_(0)^(a)f(a-x)dx]`
`rArr I=int _(0)^(pi)(e^(-cosx))/(e^(-cosx)+e^(cosx))dx` . . . (ii)
On adding Eqs . (i) and (ii) , we get
`=int_(0)^(pi)(e^(cosx)+e^(-cosx))/(e^(cosx)+e^(-cosx))dx = int_(0)^(pi)1 dx = [x] _(0)^(pi)=pi` `rArr I=pi//2`
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